Question

In Problems 19–26, apply the methods of this and the previous section to graph each function. Be sure to label key points and show at least two periods.

$y=\frac{1}{2}cot(2x-\mathrm{\pi })$

Short answer

The graph of the function $y=\frac{1}{2}cot(2x-\mathrm{\pi })$is:

Step-by-step solution

Step 1. Given

The function$y=\frac{1}{2}cot(2x-\mathrm{\pi })$

To graph the function with at least two cycles.

Step 2. Find the amplitude, period and phase shift.

Compare the given function $y=\frac{1}{2}cot\left(2\right(x-\frac{\mathrm{\pi }}{2}\left)\right)$with $y=A\sin (\omega x-\varphi )+B$

No Amplitude

Period, $T=\frac{2\mathrm{\pi }}{\omega }$

$=\frac{\mathrm{\pi }}{2}$

Phase shift, $\frac{\varphi }{\omega }=\frac{{\displaystyle \mathrm{\pi }}}{2}$

Step 3. Determine x-coordinates

The graph of $y=\frac{1}{2}cot\left(2\right(x-\frac{\mathrm{\pi }}{2}\left)\right)$

One cycle begins at $x=\frac{\varphi }{\omega }\left(\frac{\mathrm{\pi }}{2}\right)$and ends at $x=\frac{\varphi }{\omega }+\frac{2\mathrm{\pi }}{\omega }$

$$\begin{gathered} =\frac{\mathrm{\pi }}{2}+\frac{\mathrm{\pi }}{2} \\ =\mathrm{\pi } \end{gathered}$$

To find the five key points, divide the interval $(\frac{\mathrm{\pi }}{2},\mathrm{\pi })$ into four sub intervals, each of length $\frac{\mathrm{\pi }}{2}÷4=\frac{\mathrm{\pi }}{8}$

$\frac{\mathrm{\pi }}{2}+\frac{\mathrm{\pi }}{8}=\frac{5\mathrm{\pi }}{8}$

$\frac{5\mathrm{\pi }}{8}+\frac{\mathrm{\pi }}{8}=\frac{3\mathrm{\pi }}{4}$

$\frac{3\mathrm{\pi }}{4}+\frac{\mathrm{\pi }}{8}=\frac{7\mathrm{\pi }}{8}$

$\frac{7\mathrm{\pi }}{8}+\frac{\mathrm{\pi }}{8}=\mathrm{\pi }$

The $x-$coordinates are $\frac{\mathrm{\pi }}{2},\frac{5\mathrm{\pi }}{8},\frac{3\mathrm{\pi }}{4},\frac{7\mathrm{\pi }}{8},\mathrm{\pi }$

Step 4. Determine the key points.

Use these values of $x$to determine the key points on the graph:

The key points include $(\frac{5\mathrm{\pi }}{4},0),(\frac{3\mathrm{\pi }}{4},0),(\frac{7\mathrm{\pi }}{8},-\frac{1}{2}),(\frac{\mathrm{\pi }}{8},\frac{1}{2}),(\frac{\mathrm{\pi }}{4},0)$

Step 5. Sketch the graph.

Plot these five points and fill in the graph of the function.